On scale-free and poly-scale behaviors of random hierarchical networksстатья
Статья опубликована в высокорейтинговом журнале
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Дата последнего поиска статьи во внешних источниках: 26 июня 2014 г.
Аннотация:In this paper the question of statistical properties of block-hierarchical random matrices is raised for the first time in connection with structural characteristics of random hierarchical networks obtained by a ‘mipmapping’ procedure. In particular, we compute numerically the spectral density of large random adjacency matrices de fined via a hierarchy of the Bernoulli distributions q(1), q(2), ... on matrix elements, where q(gamma) depends on the hierarchy level gamma as q(gamma) = p(-mu gamma) (mu > 0). For the spectral density we clearly see scale-free behavior. We show also that for the Gaussian distributions on matrix elements with zero mean and variances sigma(gamma) = p(-nu gamma) the tail of the spectral density, rho(G)(lambda), behaves as rho(G)(lambda) similar to vertical bar lambda vertical bar(-(2-nu)/(1-nu)) for vertical bar lambda vertical bar -> infinity and 0 < nu < 1, while for nu >= 1 the power-law behavior is terminated. We also find that the vertex degree distribution of such hierarchical networks has a poly-scale fractal behavior extended over a very broad range of scales.